Enumeration of Optimal Ternary Constant-Composition Codes

Bogdanova, Galina; Kapralov, Stoyan

doi:10.1023/b:prit.0000011273.98799.a8

Cited by 14 publications

(29 citation statements)

References 4 publications

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“…The following frame starters with skew adders are in Z 4n \ {0, n, 2n, 3n}. 19,16,3,18,2,4,5,27,13,17, 22} 5,22,3,13,7,1,21,20,16,37,38,29,27,46,39,40,33,30,23, 34, 6}.…”

Section: Proposition 33mentioning

confidence: 99%

“…There exists a 4-* IGDD(6 (n+u,u) ) for each pair (n, u) ∈ {(15, 2), (17,2), (17,3), (17,4), (17,5), (17,6), (19,2), (19,3), (19,4), (…”

Section: Proposition 46mentioning

confidence: 99%

“…). If v ≡ 3, 5, 7 (mod 8) is a prime power, and v ≥ 19, 2n + 1 ≡ 1 (mod 4) is a prime, 2n + 1 > 13, then there exist both a GS(2, 4, 2vn + 1, 3) and a GS (2,4, v(2n + 1), 3).…”

Section: Lemma 62 ([30]mentioning

confidence: 99%

“…(∞, 0),(3,2),(7,4), (8, 1)} {(0, 0),(11,1),(4,4),(7,4)} {(0, 0),(15,1),(4,5),(12, 0)} {(0, 0), (1, 2), (4, 0), (3, 5)} {(0, 0), (1, 0), (16, 2), (8, 0)} {(0, 0), (3, 3), (12, 5), (10, 4)} {(0, 0), (2, 0), (5, 2), (13, 5)} {(0, 6), (12, 2), (6, 2), (5, 3)} {(∞, 0), (14, 3), (8, 0), (16, 5)} {(0, 0), (4, 1), (6, 3), (9, 1)} {(∞, 0), (18, 5), (9, 3), (16, 0)} {(0, 0), (9, 5), (14, 2), (13, 2)} {(0, 0), (1, 1), (13, 1), (16, 1)} (3) n = 22 {(0, 0), (11, 1), (13, 0), (12, 4)} {(0, 0), (4, 2), (1, 1), (12, 5)} {(∞, 0), (11, 0), (19, 2), (14, 4)} {(0, 0), (15, 4), (18, 0), (7, 3)} {(∞, 0), (9, 3), (18, 1), (16, 5)} {(0, 0), (5, 2), (12, 3), (9, 0)} {(0, 0), (16, 0), (2, 0), (18, 1)} { (0, 0), (8, 4), (4, 1), (2, 3)} { (0, 0), (14, 1), (20, 1), (10, 1)} {(0, 0), (1, 0), (7, 4), (13, 1)} {(0, 0), (1, 4), (5, 3), (16, 5)}.…”

mentioning

confidence: 99%

“…21,2),(21,3),(21,5),(21,6),(23,2),(23,4),(23,5),(25,2),(25,5),(27,2),(29,2),(31,3),(31,4),(33,2), (35, 2), (35, 3), (35, 4), (47, 2)}.…”

mentioning

confidence: 99%

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4‐*GDD(6ⁿ)s and Related Optimal Quaternary Constant‐Weight Codes

Zhu¹,

Ge²

2012

J of Combinatorial Designs

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Constant‐weight codes (CWCs) have played an important role in coding theory. To construct CWCs, a K‐GDD (where GDD is group divisible design) with the “star” property, denoted by K‐*GDD, was introduced, in which any two intersecting blocks intersect in at most two common groups. In this paper, we consider the existence of 4‐*GDD(gn)s. Previously, the necessary conditions for existence were shown to be sufficient for g=3, and also sufficient for g=6 with prime powers n≡3,5,7(mod8) and n≥19. We continue to investigate the existence of 4‐*GDD(6n)s and show that the necessary condition for the existence of a 4‐*GDD(6n), namely, n≥14, is also sufficient. The known results on the existence of optimal quaternary (n, 5, 4) CWCs are also extended.

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“…The following frame starters with skew adders are in Z 4n \ {0, n, 2n, 3n}. 19,16,3,18,2,4,5,27,13,17, 22} 5,22,3,13,7,1,21,20,16,37,38,29,27,46,39,40,33,30,23, 34, 6}.…”

Section: Proposition 33mentioning

confidence: 99%

“…There exists a 4-* IGDD(6 (n+u,u) ) for each pair (n, u) ∈ {(15, 2), (17,2), (17,3), (17,4), (17,5), (17,6), (19,2), (19,3), (19,4), (…”

Section: Proposition 46mentioning

confidence: 99%

“…). If v ≡ 3, 5, 7 (mod 8) is a prime power, and v ≥ 19, 2n + 1 ≡ 1 (mod 4) is a prime, 2n + 1 > 13, then there exist both a GS(2, 4, 2vn + 1, 3) and a GS (2,4, v(2n + 1), 3).…”

Section: Lemma 62 ([30]mentioning

confidence: 99%

mentioning

confidence: 99%

“…21,2),(21,3),(21,5),(21,6),(23,2),(23,4),(23,5),(25,2),(25,5),(27,2),(29,2),(31,3),(31,4),(33,2), (35, 2), (35, 3), (35, 4), (47, 2)}.…”

mentioning

confidence: 99%

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4‐*GDD(6ⁿ)s and Related Optimal Quaternary Constant‐Weight Codes

Zhu¹,

Ge²

2012

J of Combinatorial Designs

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Optimal ternary constant-composition codes with weight four and distance six

Wei

Zhang

Zhu

et al. 2015

Discrete Mathematics

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The sizes of optimal constant-composition codes of weight three have been determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes played an important role in their constructions. In this paper, we study the problem of constructing optimal ternary constant-composition codes with Hamming weight four and minimum distance six. The problem is solved with a small number of lengths undetermined. The previously known results are those with code length no greater than 10.

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